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Classical product code constructions for quantum Calderbank-Shor-Steane codes

Dimiter Ostrev, Davide Orsucci, Francisco Lázaro, Balazs Matuz

TL;DR

This work introduces a new class of quantum CSS codes derived from classical product codes, enabling both X and Z parity checks to be built from classical product or tensor product codes while automatically satisfying the CSS commutativity constraint. The framework yields flexible asymmetric and symmetric 2‑fold constructions and a multi‑fold generalization, including a SPC(D) specialization that achieves high distance with reasonable rate. A standout result is a 3‑fold SPC code with parameters [[512,174,8]] that demonstrates strong performance under erasure decoding and belief‑propagation decoding for depolarising noise, outperforming several well‑known quantum LDPC families in the same regime. The authors also develop meta‑checks and an extended decoding scheme to correct syndrome readout errors, showing robustness to measurement faults and outlining practical routes toward fault‑tolerant operation. Overall, the approach blends classical coding techniques with quantum stabilizer requirements to yield high‑rate, moderately large‑distance CSS codes with favorable decoding properties and hardware‑friendly syndrome measurements, offering a promising path toward near‑term experimental implementations.

Abstract

Several notions of code products are known in quantum error correction, such as hyper-graph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalisation of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both $X$ parity checks and $Z$ parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialise to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC $3$-fold product CSS code having parameters $[[512,174,8]]$ are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarising noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

Classical product code constructions for quantum Calderbank-Shor-Steane codes

TL;DR

This work introduces a new class of quantum CSS codes derived from classical product codes, enabling both X and Z parity checks to be built from classical product or tensor product codes while automatically satisfying the CSS commutativity constraint. The framework yields flexible asymmetric and symmetric 2‑fold constructions and a multi‑fold generalization, including a SPC(D) specialization that achieves high distance with reasonable rate. A standout result is a 3‑fold SPC code with parameters [[512,174,8]] that demonstrates strong performance under erasure decoding and belief‑propagation decoding for depolarising noise, outperforming several well‑known quantum LDPC families in the same regime. The authors also develop meta‑checks and an extended decoding scheme to correct syndrome readout errors, showing robustness to measurement faults and outlining practical routes toward fault‑tolerant operation. Overall, the approach blends classical coding techniques with quantum stabilizer requirements to yield high‑rate, moderately large‑distance CSS codes with favorable decoding properties and hardware‑friendly syndrome measurements, offering a promising path toward near‑term experimental implementations.

Abstract

Several notions of code products are known in quantum error correction, such as hyper-graph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalisation of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both parity checks and parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialise to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC -fold product CSS code having parameters are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarising noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.
Paper Structure (42 sections, 54 equations, 4 figures, 1 table)

This paper contains 42 sections, 54 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Comparison with other product code constructions
  3. Classical product construction for CSS codes
  4. Preliminaries and notation
  5. Classical codes
  6. Classical product codes ($\mathcal{C}_\times$)
  7. Classical binary tensor product codes ($\mathcal{C}_\otimes$)
  8. Quantum CSS codes
  9. Main constructions
  10. Asymmetric $2$-fold product construction of CSS codes ($\mathcal{C}_\ltimes$)
  11. Symmetric 2-fold product construction of CSS codes ($\mathcal{C}_\boxtimes$)
  12. Symmetric $\textbf{D}$-fold product construction of CSS codes ($\mathcal{C}_\textbf{(D)}$)
  13. Single-parity-check $\textbf{D}$-fold product CSS codes ($\mathcal{C}_{\textbf{SPC(D)}}$)
  14. Comparison with previous works
  15. Previous code families participating in the comparison
  16. ...and 27 more sections

Figures (4)

  • Figure 1: Array representation of the $54$ bits of a code given by the product of a $[9, 6]$ and of a $[6, 4]$ systematic code. The final rate is $R = (6/9) (4/6) = 4/9$. The bits on white cells correspond to message bits, bits on hatched cells corresponds to parity-check bits, bits on doubly-hatched cells to checks-of-checks or meta-checks.
  • Figure 2: Monte Carlo simulation of the logical error rates of 5 different CSS codes. Left: maximum likelihood decoding under erasure channel ($\mathcal{E}_\beta$). Right: BP decoding under depolarising channel ($\mathcal{D}_\epsilon$). See main text for details.
  • Figure 3: Monte Carlo simulations of the logical error rates under BP decoding under a depolarizing channel with syndrome measurement errors. Left: all codes are subject to either no syndrome readout error (solid lines) or to a syndrome readout flip with probability $p=10^{-3}$ (dashed lines). Right: performance of the SPC(3) code for different values of the syndrome readout error probability $p$.
  • Figure 4: Example binary/quaternary inter-conversion in message passing for a quantum Tanner graph having 7 VNs (qubits) and 3 CNs (stabiliser generators). Edges transmitting messages to one CN $\mathtt{c}_j$ (top) and to one VN $\mathtt{v}_i$ (bottom) are highlighted and the edges are labelled (coloured) according to the Pauli operator measured by the stabiliser $S_i$ on qubit $j$.

Theorems & Definitions (8)

  • Remark
  • Definition 1
  • Definition 2
  • Remark
  • Definition 3
  • Definition 4
  • Remark
  • Remark